THIN STATIONARY SETS AND DISJOINT CLUB SEQUENCES

SY-DAVID FRIEDMAN AND JOHN KRUEGER

Abstract. We describe two opposing combinatorial properties related to adding clubs to ω_{2}: the existence of a thin stationary subset of P_{ω_{1}}(ω_{2}) and the existence of a disjoint club sequence on ω_{2}. A special Aronszajn tree on ω_{2} implies there exists a thin stationary set. If there exists a disjoint club sequence, then there is no thin stationary set, and moreover there is a fat stationary subset of ω_{2} which cannot acquire a club subset by any forcing poset which preserves ω_{1} and ω_{2}. We prove that the existence of a disjoint club sequence follows from Martin’s Maximum and is equiconsistent with a Mahlo cardinal.

Suppose that S is a fat stationary subset of ω_{2}, that is, for every club set C ⊆ ω_{2}, S ∩ C contains a closed subset with order type ω_{1} + 1. A number of forcing posets have been defined which add a club subset to S and preserve cardinals under various assumptions. Abraham and Shelah [1] proved that, assuming CH, the poset consisting of closed bounded subsets of S ordered by end-extension adds a club subset to S and is ω_{1}-distributive. S. Friedman [5] discovered a different poset for adding a club subset to a fat set S ⊆ ω_{2} with finite conditions ^{1}. This finite club poset preserves all cardinals provided that there exists a thin stationary subset of P_{ω_{1}}(ω_{2}), that is, a stationary set T ⊆ P_{ω_{1}}(ω_{2}) such that for all β < ω_{2}, |{a∩β : a ∈ T }| ≤ ω_{1}. This notion of stationarity appears in [9] and was discovered independently by Friedman. The question remained whether it is always possible to add a club subset to a given fat set and preserve cardinals, without any assumptions.

J. Krueger introduced a combinatorial principle on ω_{2} which asserts the existence of a disjoint club sequence, which is a pairwise disjoint sequence ⟨C_{α} : α ∈ A⟩ indexed by a stationary subset of ω_{2} ∩ cof(ω_{1}), where each C_{α} is club in P_{ω_{1}}(α). Krueger proved that the existence of such a sequence implies there is a fat stationary set S ⊆ ω_{2} which cannot acquire a club subset by any forcing poset which preserves ω_{1} and ω_{2}.

We prove that a special Aronszajn tree on ω_{2} implies there exists a thin stationary subset of P_{ω_{1}}(ω_{2}). On the other hand assuming Martin’s Maximum there exists a disjoint club sequence on ω_{2}. Moreover, we have the following equiconsistency result.

Theorem 0.1. Each of the following statements is equiconsistent with a Mahlo cardinal: (1) There does not exist a thin stationary subset of P_{ω_{1}}(ω_{2}). (2) There exists a disjoint club sequence on ω_{2}. (3) There exists a fat stationary set S ⊆ ω_{2} such that any forcing poset which preserves ω_{1} and ω_{2} does not add a club subset to S.

Date: June 2005.

2000 Mathematics Subject Classification. 03E35; 03E40.

Both authors were supported by FWF project number P16790-N04.

1A similar poset was defined independently by Mitchell [7]

Our proof of this theorem gives a totally different construction of the following result of Mitchell [8]: If κ is Mahlo in L, then there is a generic extension of L in which κ = ω_{2} and there is no special Aronszajn tree on ω_{2}. The consistency of Theorem 0.1(3) provides a negative solution to the following problem of Abraham and Shelah [1]: if S ⊆ ω_{2} is fat, does there exist an ω_{1}-distributive forcing poset which adds a club subset to S?

Section 1 outlines notation and background material. In Section 2 we discuss thin stationarity and prove that a special Aronszajn tree implies the existence of a thin stationary set. In Section 3 we introduce disjoint club sequences and prove that the existence of such a sequence implies there is a fat stationary set in ω_{2} which cannot acquire a club subset by any forcing poset which preserves ω_{1} and ω_{2}. In Section 4 we prove that Martin’s Maximum implies there exists a disjoint club sequence. In Section 5 we construct a model in which there is a disjoint club sequence using an RCS iteration up to a Mahlo cardinal.

Sections 3 and 4 are due for the most part to J. Krueger. We would like to thank Boban Veličković and Mirna Dzamonja for pointing out Theorem 2.3 to the authors.

1. Preliminaries

For a set X which contains ω_{1}, P_{ω_{1}}(X) denotes the collection of countable subsets of X. A set C ⊆ P_{ω_{1}}(X) is club if it is closed under unions of countable increasing sequences and is cofinal. A set S ⊆ P_{ω_{1}}(X) is stationary if it meets every club. If C ⊆ P_{ω_{1}}(X) is club then there exists a function F : X^{<ω} → X such that every a in P_{ω_{1}}(X) closed under F is in C. If F : X^{<ω} → P_{ω_{1}}(X) is a function and Y ⊆ X, we say that Y is closed under F if for all γ⃗ from Y^{<ω}, F (γ⃗) ⊆ Y. A partial function H : P_{ω_{1}}(X) → X is regressive if for all a in the domain of H, H(a) is a member of a. Fodor’s Lemma asserts that whenever S ⊆ P_{ω_{1}}(X) is stationary and H : S → X is a total regressive function, there is a stationary set S^{∗} ⊆ S and a set x in X such that for all a in S^{∗}, H(a) = x.

If κ is a regular cardinal let cof(κ) (respectively cof(< κ)) denote the class of ordinals with cofinality κ (respectively cofinality less than κ). If A is a cofinal subset of a cardinal λ and κ < λ, we write for example A ∪ cof(κ) to abbreviate A ∪ (λ ∩ cof(κ)).

A stationary set S ⊆ κ is fat if for every club C ⊆ κ, S ∩ C contains closed subsets with arbitrarily large order types less than κ. If κ is the successor of a regular uncountable cardinal μ, this is equivalent to the statement that for every club C ⊆ κ, S ∩ C contains a closed subset with order type μ + 1. In particular, if A ⊆ κ^{+} ∩ cof(μ) is stationary then A ∪ cof(< μ) is fat.

We write θ ≫ κ to indicate θ is larger than 2^{2^{|H(κ)|}}.

A tree T is a special Aronszajn tree on ω_{2} if:

(1) T has height ω_{2} and each level has size less than ω_{2},

(2) each node in T is an injective function f : α → ω_{1} for some α < ω_{2},

(3) the ordering on T is by extension of functions, and if f is in T then f ↾ β is in T for all β < dom(f).

By [8] if there does not exist a special Aronszajn tree on ω_{2}, then ω_{2} is a Mahlo cardinal in L.

If V is a transitive model of ZFC, we say that W is an outer model of V if W is a transitive model of ZFC such that V ⊆ W and W has the same ordinals as V.

A forcing poset ℙ is κ-distributive if forcing with ℙ does not add any new sets of ordinals with size κ.

If ℙ is a forcing poset, ȧ is a ℙ-name, and G is a generic filter for ℙ, we write a for the set ȧ^{G}.

Martin’s Maximum is the statement that whenever ℙ is a forcing poset which preserves stationary subsets of ω_{1}, then for any collection D of dense subsets of ℙ with |D| ≤ ω_{1}, there is a filter G ⊆ ℙ which intersects each dense set in D.

A forcing poset ℙ is proper if for all sufficiently large regular cardinals θ > 2^{|ℙ|}, there is a club of countable elementary substructures N of ⟨H(θ), ∈⟩ such that for all p in N ∩ℙ, there is q ≤ p which is generic for N, i.e. q forces N[Ġ]∩On = N ∩On. If ℙ is proper then ℙ preserves ω_{1} and preserves stationary subsets of P_{ω_{1}}(λ) for all λ ≥ ω_{1}. A forcing poset ℙ is semiproper if the same statement holds as above except the requirement that q is generic is replaced by q being semigeneric, i.e. q forces N[Ġ] ∩ ω_{1} = N ∩ ω_{1}. If ℙ is semigeneric then ℙ preserves ω_{1} and preserves stationary subsets of ω_{1}.

If ℙ is ω_{1}-c.c. and N is a countable elementary substructure of H(θ), then ℙ forces N[Ġ] ∩ On = N ∩ On; so every condition in ℙ is generic for N.

We let ^{<ω}On denote the class of finite strictly increasing sequences of ordinals. If η and ν are in ^{<ω}On, write η ⊴ ν if η is an initial segment of ν, and write η ◁ ν if η ⊴ ν and η ≠ ν. Let l(η) denote the length of η. A set T ⊆ ^{<ω}On is a tree if for all η in T and k < l(η), η ↾ k is in T. A cofinal branch of T is a function b : ω → κ such that for all n < ω, b ↾ n is in T.

Suppose I is an ideal on a set X. Then I^{+} is the collection of subsets of X which are not in I. If S is in I^{+} let I ↾ S denote the ideal I ∩ P(S). For example if I = NS_{κ}, the ideal of non-stationary subsets of κ, a set S is in I^{+} iff S is stationary. In this case NS_{κ} ↾ S is the ideal of non-stationary subsets of S and (NS_{κ} ↾ S)^{+} is the collection of stationary subsets of S.

If κ is regular and λ ≥ κ is a cardinal, then Coll(κ, λ) is a forcing poset for collapsing λ to have cardinality κ: conditions are partial functions p : κ → λ with size less than κ, ordered by extension of functions.

2. Thin Stationary Sets

Let T be a cofinal subset of P_{ω_{1}}(ω_{2}). We say that T is thin if for all β < ω_{2} the set {a ∩ β : a ∈ T } has size less than ω_{2}. Note that if CH holds then P_{ω_{1}}(ω_{2}) itself is thin. A set S ⊆ P_{ω_{1}}(ω_{2}) is closed under initial segments if for all a in S and β < ω_{2}, a ∩ β is in S.

Lemma 2.1. If S ⊆ P_{ω_{1}}(ω_{2}) is stationary and closed under initial segments, then for all uncountable β < ω_{2}, the set S ∩ P_{ω_{1}}(β) is stationary in P_{ω_{1}}(β).

Proof. Consider β < ω_{2} and let C ⊆ P_{ω_{1}}(β) be a club set. Then the set D = {a ∈ P_{ω_{1}}(ω_{2}) : a ∩ β ∈ C} is a club subset of P_{ω_{1}}(ω_{2}). Fix a in S ∩ D. Since S is closed under initial segments, a ∩ β is in S ∩ C. □

Lemma 2.2. If there exists a thin stationary subset of P_{ω_{1}}(ω_{2}), then there is a thin stationary set S such that for all uncountable β < ω_{2}, S ∩ P_{ω_{1}}(β) is stationary in P_{ω_{1}}(β).

Proof. Let T be a thin stationary set. Define S = {a ∩ β : a ∈ T, β < ω_{2}}. Then S is thin stationary and closed under initial segments. □

A set S ⊆ P_{ω_{1}}(ω_{2}) is a local club if there is a club set C ⊆ ω_{2} such that for all uncountable α in C, S ∩ P_{ω_{1}}(α) contains a club in P_{ω_{1}}(α) (see [3]). Note that local clubs are stationary.

Theorem 2.3. If there is a special Aronszajn tree on ω_{2}, then there is a thin local club subset of P_{ω_{1}}(ω_{2}).

Proof. Let T be a special Aronszajn tree on ω_{2}. For each f in T with dom(f) ≥ ω_{1}, define S_{f} = {{α ∈ dom(f) : f(α) < i} : i < ω_{1}}. Note that S_{f} is a club subset of P_{ω_{1}}(domf). For each uncountable β < ω_{2} define S_{β} = ⋃{S_{f} : f ∈ T , dom(f) = β}. Then S_{β} has size ω_{1}. Now define S = ⋃{S_{β} : ω_{1} ≤ β < ω_{2}}. Clearly S is a local club. To show S is thin, it suffices to prove that whenever β < γ are uncountable and a is in S_{γ}, then a∩β is in S_{β}. Fix f in T and i < ω_{1} such that a = f^{−1}“i. Then f ↾ β is in T , so (f ↾ β)^{−1}“i is in S_{β}. But (f ↾ β)^{−1}“i = (f^{−1}“i) ∩ β = a ∩ β. □

In later sections of the paper we will construct models in which there does not exist a thin stationary subset of P_{ω_{1}}(ω_{2}). Theorem 2.3 shows that in such a model there cannot exist a special Aronszajn tree on ω_{2}, so by [8] ω_{2} is Mahlo in L. Mitchell [8] constructed a model in which there is no special Aronszajn tree on ω_{2} by collapsing a Mahlo cardinal in L to become ω_{2} with a proper forcing poset. However, in Mitchell’s model the set (P_{ω_{1}}(κ))^{L} is a thin stationary subset of P_{ω_{1}}(ω_{2}).

Lemma 2.4. Suppose S ⊆ P_{ω_{1}}(ω_{2}) is a local club. Then S is a local club in any outer model W with the same ω_{1} and ω_{2}.

Proof. Let C be a club subset of ω_{2} such that for every uncountable α in C, S ∩ P_{ω_{1}}(α) contains a club in P_{ω_{1}}(α). Then C remains club in W. For each uncountable α in C, fix a bijection g_{α} : ω_{1} → α. Then {g_{α}“i : i < ω_{1}} is a club subset of P_{ω_{1}}(α). By intersecting this club with S, we get a club subset of S ∩ P_{ω_{1}}(α) of the form {a_{i}^{α} : i < ω_{1}} which is increasing and continuous. Clearly this set remains a club subset of P_{ω_{1}}(α) in W. □

Proposition 2.5. (1) Suppose there exists a thin local club in P_{ω_{1}}(ω_{2}). Then there exists a thin local club in any outer model with the same ω_{1} and ω_{2}. (2) Suppose κ is a cardinal such that for all μ < κ, μ^{ω} < κ, and assume ℙ is a proper forcing poset which collapses κ to become ω_{2}. Then ℙ forces there is a thin stationary subset of P_{ω_{1}}(ω_{2}).

Proof. (1) is immediate from Lemma 2.4 and the absoluteness of thinness. (2) Let G be generic for ℙ over V and work in V [G]. Since ℙ is proper, ω_{1} is preserved and the set S = (P_{ω_{1}}(κ))^{V} is stationary in P_{ω_{1}}(ω_{2}). We claim that S is thin. If β < ω_{2} then {a ∩ β : a ∈ S} = (P_{ω_{1}}(β))^{V}, and |(P_{ω_{1}}(β))^{V} | ≤ |(P_{ω_{1}}(β))^{V} |^{V} = |(β^{ω})^{V} |^{V} < κ. □

As we mentioned above, if CH holds then the set P_{ω_{1}}(ω_{2}) itself is thin. We show on the other hand that if CH fails then no club subset of P_{ω_{1}}(ω_{2}) is thin. The proof is due to Baumgartner and Taylor [2] who showed that for any club set C ⊆ P_{ω_{1}}(ω_{2}), there is a countable set A ⊆ ω_{2} such that C ∩ P(A) has size at least 2^{ω}. Their method of proof is described in the next lemma; we include the proof since we will use similar arguments later in the paper.

Lemma 2.6. Suppose Z is a stationary subset of ω_{2} ∩ cof(ω) and for each α in Z, M_{α} is a countable cofinal subset of α. Then there is a sequence ⟨Z_{s}, ξ_{s} : s ∈ ^{<ω}2⟩ satisfying:

(1) each Z_{s} is a stationary subset of Z,

(2) if s ⊴ t then Z_{t} ⊆ Z_{s},

(3) if α is in Z_{s} then ξ_{s} is in M_{α},

(4) if α is in Z_{s 0} and β is in Z_{s 1}, then ξ_{s 0} is not in M_{β} and ξ_{s 1} is not in M_{α}.

Proof. Let Z_{⟨⟩} = Z and let ξ_{⟨⟩} = 0. Suppose Z_{s} is given. Define X_{s} as the set of ξ in ω_{2} such that the set {α ∈ Z_{s} : ξ ∈ M_{α}} is stationary. A straightforward argument using Fodor’s Lemma shows that X_{s} is unbounded in ω_{2}. For each α in Z_{s} such that X_{s} ∩ α has size ω_{1}, there exists ξ < α in X_{s} such that ξ is not in M_{α}. By Fodor’s Lemma there is a stationary set Z′_{s 1} ⊆ Z_{s} and ξ_{s 0} in X_{s} such that for all α in Z′_{s 1}, ξ_{s 0} is not in M_{α}. Let Z′_{s 0} denote the set of α in Z_{s} such that ξ_{s 0} is in M_{α}, which is stationary since ξ_{s 0} is in X_{s}. Now define Y_{s} as the set of ξ in ω_{2} such that {α ∈ Z′_{s 1} : ξ ∈ M_{α}} is stationary. Then Y_{s} is unbounded in ω_{2}. So for each α in Z′_{s 0} such that Y_{s} ∩ α has size ω_{1}, there is ξ < α in Y_{s} which is not in M_{α}. By Fodor’s Lemma there is ξ_{s 1} in Y_{s} and Z_{s 0} ⊆ Z′_{s 0} stationary such that for all α in Z_{s 0}, ξ_{s 1} is not in M_{α}. Now define Z_{s 1} as the set of α in Z′_{s 1} such that ξ_{s 1} is in M_{α}. □

Theorem 2.7 (Baumgartner and Taylor). If C ⊆ P_{ω_{1}}(ω_{2}) is club, then there is a countable set A ⊆ ω_{2} such that C ∩ P(A) has size at least 2^{ω}. Hence if CH fails then there does not exist a thin club subset of P_{ω_{1}}(ω_{2}).

Proof. Let F : ω_{2}^{<ω} → ω_{2} be a function such that any a in P_{ω_{1}}(ω_{2}) closed under F is in C. Let Z be the stationary set of α in ω_{2} ∩ cof(ω) closed under F. For each α in Z fix a countable set M_{α} ⊆ α such that sup(M_{α}) = α and M_{α} is closed under F. Fix a sequence ⟨Z_{s}, ξ_{s} : s ∈ ^{<ω}2} as described in Lemma 2.6.

For each function f : ω → 2 define b_{f} = cl_{F} ({ξ_{f↾n} : n < ω}). Then b_{f} is in C. Note that if n < ω and α is in Z_{f↾n}, then cl_{F} ({ξ_{f↾m} : m ≤ n}) ⊆ M_{α}. For by Lemma 2.6(2), for m ≤ n, Z_{f↾n} ⊆ Z_{f↾m}. So α is in Z_{f↾m}, and hence ξ_{f↾m} is in M_{α} by (3). But M_{α} is closed under F.

Let A = cl_{F} ({ξ_{s} : s ∈ ^{<ω}2}). Since ^{<ω}2 has size ω, A is countable, and clearly each b_{f} is a subset of A. We claim that for distinct f and g, b_{f} ≠ b_{g}. Let n < ω be least such that f(n) ≠ g(n). If b_{f} = b_{g}, then there is k > n such that ξ_{g↾(n+1)} is in cl_{F} ({ξ_{f↾m} : m ≤ k}). Fix α in Z_{f↾k}. By the last paragraph, ξ_{g↾(n+1)} is in M_{α}. But α is in Z_{f↾(n+1)} by (2), which contradicts (4). □

Let κ be an uncountable cardinal. The Weak Reflection Principle at κ is the statement that whenever S is a stationary subset of P_{ω_{1}}(κ), there is a set Y in P_{ω_{2}}(κ) such that ω_{1} ⊆ Y and S ∩ P_{ω_{1}}(Y) is stationary in P_{ω_{1}}(Y). Martin’s Maximum implies the Weak Reflection Principle holds for all uncountable cardinals κ [4]. The Weak Reflection Principle at ω_{2} is equivalent to the statement that for every stationary set S ⊆ P_{ω_{1}}(ω_{2}), there is a stationary set of uncountable β < ω_{2} such that S ∩ P_{ω_{1}}(β) is stationary in P_{ω_{1}}(β). This is equivalent to the statement that every local club subset of P_{ω_{1}}(ω_{2}) contains a club. The Weak Reflection Principle at ω_{2} is equiconsistent with a weakly compact cardinal [3].

Corollary 2.8. Suppose CH fails and there is a special Aronszajn tree on ω_{2}. Then the Weak Reflection Principle at ω_{2} fails.

Proof. By Theorems 2.3 and 2.7, there is a thin local club subset of P_{ω_{1}}(ω_{2}) which is not club. Hence the Weak Reflection Principle at ω_{2} fails. □

In Sections 4 and 5 we describe models in which there is no thin stationary subset of P_{ω_{1}}(ω_{2}). On the other hand S. Friedman proved there always exists a thin cofinal set.

Theorem 2.9 (Friedman). There exists a thin cofinal subset of P_{ω_{1}}(ω_{2}).

Proof. We construct by induction a sequence ⟨S_{α} : ω_{1} ≤ α < ω_{2}⟩ satisfying the properties: (1) each S_{α} is a cofinal subset of P_{ω_{1}}(α) with size ω_{1}, (2) for uncountable β < γ, if a is in S_{γ} then a ∩ β is in ⋃{S_{α} : ω_{1} ≤ α ≤ β}, and (3) if β < γ < ω_{2}, a is in P_{ω_{1}}(γ), and a∩β is in S_{β}, then there is b in S_{γ} such that a ⊆ b and a∩β = b∩β.

Let S_{ω_{1}} = ω_{1}. Given S_{α}, let S_{α+1} be the collection {b∪{α} : b ∈ S_{α}}. Conditions (1), (2), and (3) follow by induction. Suppose γ < ω_{2} is an uncountable limit ordinal and S_{α} is defined for all uncountable α < γ. If cf(γ) = ω_{1} then let S_{γ} = ⋃{S_{α} : ω_{1} ≤ α < γ}. The required conditions follow by induction.

Assume cf(γ) = ω. Fix an increasing sequence of uncountable ordinals ⟨γ_{n} : n < ω⟩ unbounded in γ. Let T_{γ} be some cofinal subset of P_{ω_{1}}(γ) with size ω_{1}. Fix n < ω. For each x in T_{γ} and a in S_{γ_{n}} define a set b(a, x, n) in P_{ω_{1}}(γ) inductively as follows. Let b(a, x, n) ∩ γ_{n} = a. Given b(a, x, n) ∩ γ_{m} in S_{γ_{m}} for some m ≥ n, apply condition (3) to γ_{m}, γ_{m+1}, and the set

(b(a, x, n) ∩ γ_{m}) ∪ ((x ∩ [γ_{m}, γ_{m+1}))

to find y in S_{γ_{m+1}} such that y ∩ γ_{m} = b(a, x, n) ∩ γ_{m} and x ∩ [γ_{m}, γ_{m+1}) ⊆ y. Let b(a, x, n) ∩ γ_{m+1} = y. This completes the definition of b(a, x, n). Clearly b(a, x, n) ∩ γ_{n} = a, x \ γ_{n} ⊆ b(a, x, n), and for all k ≥ n, b(a, x, n) ∩ γ_{k} is in S_{γ_{k}}.

Now define S_{γ} = {b(a, x, n) : n < ω, a ∈ S_{γ_{n}}, x ∈ T_{γ}}. We verify conditions (1), (2), and (3). Clearly S_{γ} has size ω_{1}. Let β < γ and consider b(a, x, n) in S_{γ}. Fix k > n such that β < γ_{k}. Then b(a, x, n) ∩ γ_{k} is in S_{γ_{k}}. So by induction b(a, x, n) ∩ β is in ⋃{S_{α} : ω_{1} ≤ α ≤ β}. Now assume a is in P_{ω_{1}}(γ), β < γ, and a ∩ β is in S_{β}. Choose x in T_{γ} such that a ⊆ x. Fix k such that β < γ_{k}. By the induction hypothesis there is a′ in S_{γ_{k}} such that a ∩ γ_{k} ⊆ a′ and a′ ∩ β = a ∩ β. Let c = b(a′, x, k). Then c is in S_{γ}, c ∩ β = (c ∩ γ_{k}) ∩ β = a′ ∩ β = a ∩ β, and a ⊆ c.

To prove S_{γ} is cofinal consider a in P_{ω_{1}}(γ). Fix x in T_{γ} such that a ⊆ x. By induction S_{γ_{0}} is cofinal in P_{ω_{1}}(γ_{0}). So let y be in S_{γ_{0}} such that x ∩ γ_{0} ⊆ y. Then a is a subset of b(y, x, 0).

Now define S = ⋃{S_{β} : ω_{1} ≤ β < ω_{2}}. Conditions (1) and (2) imply that S is thin and cofinal in P_{ω_{1}}(ω_{2}). □

3. Disjoint Club Sequences

We introduce a combinatorial property of ω_{2} which implies there does not exist a thin stationary subset of P_{ω_{1}}(ω_{2}). This property follows from Martin’s Maximum and is equiconsistent with a Mahlo cardinal. It implies there exists a fat stationary subset of ω_{2} which cannot acquire a club subset by any forcing poset which preserves ω_{1} and ω_{2}.

Definition 3.1. A disjoint club sequence on ω_{2} is a sequence ⟨C_{α} : α ∈ A⟩ such that A is a stationary subset of ω_{2} ∩ cof(ω_{1}), each C_{α} is a club subset of P_{ω_{1}}(α), and C_{α} ∩ C_{β} is empty for all α < β in A.

Proposition 3.2. Suppose there is a disjoint club sequence on ω_{2}. Then there does not exist a thin stationary subset of P_{ω_{1}}(ω_{2}).

Proof. Let ⟨C_{α} : α ∈ A⟩ be a disjoint club sequence. Suppose for a contradiction there exists a thin stationary set. By Lemma 2.2 fix a thin stationary set T ⊆ P_{ω_{1}}(ω_{2}) such that for all uncountable β < ω_{2}, T ∩ P_{ω_{1}}(β) is stationary in P_{ω_{1}}(β). Then for each β in A we can choose a set a_{β} in C_{β}∩T. Since cf(β) = ω_{1}, sup(a_{β}) < β. By Fodor’s Lemma there is a stationary set B ⊆ A and a fixed γ < ω_{2} such that for all β in B, sup(a_{β}) = γ. If α < β are in B, then a_{α} ≠ a_{β} since C_{α} ∩ C_{β} is empty. So the set {a_{β} : β ∈ B} witnesses that T is not thin, which is a contradiction. □

Lemma 3.3. Suppose there is a disjoint club sequence ⟨C_{α} : α ∈ A⟩ on ω_{2}. Let W be an outer model with the same ω_{1} and ω_{2} in which A is still stationary. Then there is a disjoint club sequence ⟨D_{α} : α ∈ A⟩ in W.

Proof. By the proof of Lemma 2.4, each C_{α} contains a club set D_{α} in W. Since ω_{1} is preserved, each α in A still has cofinality ω_{1}. □

Theorem 3.4. Suppose ⟨C_{α} : α ∈ A⟩ is a disjoint club sequence on ω_{2}. Then A ∪ cof(ω) does not contain a club.

Proof. Suppose for a contradiction that A∪cof(ω) contains a club. Without loss of generality 2^{ω_{1}} = ω_{2}. Otherwise work in a generic extension W by Coll(ω_{2}, 2^{ω_{1}}): in W the set A ∪ cof(ω) contains a club and by Lemma 3.3 there is a disjoint club sequence ⟨D_{α} : α ∈ A⟩.

Since 2^{ω_{1}} = ω_{2}, H(ω_{2}) has size ω_{2}. Fix a bijection h : H(ω_{2}) → ω_{2}. Let A denote the structure ⟨H(ω_{2}), ∈, h⟩. Define B as the set of α in ω_{2} ∩ cof(ω_{1}) such that there exists an increasing and continuous sequence ⟨N_{i} : i < ω_{1}⟩ of countable elementary substructures of A such that:

(1) for i < ω_{1}, N_{i} is in N_{i+1},

(2) the set {N_{i} ∩ ω_{2} : i < ω_{1}} is club in P_{ω_{1}}(α).

We claim that B is stationary in ω_{2}. To prove this let C ⊆ ω_{2} be club. Let B be the expansion of A by the function α ↦ min(C \ α). Define by induction an increasing and continuous sequence ⟨N_{i} : i < ω_{1}⟩ of elementary substructures of B such that for all i < ω_{1}, N_{i} is in N_{i+1}. Let N = ⋃{N_{i} : i < ω_{1}}. Then ω_{1} ⊆ N so N ∩ ω_{2} is an ordinal. Write α = N ∩ ω_{2}. Then α is in C and {N_{i} ∩ ω_{2} : i < ω_{1}} is club in P_{ω_{1}}(α). So α is in B ∩ C.

Since A ∪ cof(ω) contains a club, A ∩ B is stationary. For each α in A ∩ B fix a sequence ⟨N_{i}^{α} : i < ω_{1}⟩ as described in the definition of B. Then {N_{i}^{α} ∩ ω_{2} : i < ω_{1}} ∩ C_{α} is club in P_{ω_{1}}(α). So there exists a club set c_{α} ⊆ ω_{1} such that {N_{i}^{α} ∩ ω_{2} : i ∈ c_{α}} is club and is a subset of C_{α}. Write i_{α} = min(c_{α}) and let d_{α} = c_{α} \ {i_{α}}.

Define S = {N_{i}^{α} : α ∈ A ∩ B, i ∈ d_{α}}. If N is in S then there is a unique pair α in A ∩ B and i in d_{α} such that N = N_{i}^{α}. For if N = N_{i}^{α} = N_{j}^{β}, then N ∩ ω_{2} is in C_{α} ∩ C_{β}, so α = β. Clearly then i = j. Also note that if N_{i}^{α} is in S then N_{i_{α}}^{α} is in N_{i}^{α}. So the function H : S → H(ω_{2}) defined by H(N_{i}^{α}) = N_{i_{α}}^{α} is well-defined and regressive.

We claim that S is stationary in P_{ω_{1}}(H(ω_{2})). To prove this let F : H(ω_{2})^{<ω} → H(ω_{2}) be a function. Define G : ω_{2}^{<ω} → ω_{2} by letting G(α_{0}, …, α_{n}) be equal to h(F (h^{−1}(α_{0}), …, h^{−1}(α_{n}))). Let E be the club set of α in ω_{2} closed under G. Fix α in E ∩A∩B. Then there is i in d_{α} such that N_{i}^{α} ∩ω_{2} is closed under G. We claim that N_{i}^{α} is closed under F. Given a_{0}, …, a_{n} in N_{i}^{α}, the ordinals h(a_{0}), …, h(a_{n}) are in N_{i}^{α} ∩ ω_{2}. So γ = G(h(a_{0}), …, h(a_{n})) = h(F (a_{0}, …, a_{n})) is in N_{i}^{α} ∩ ω_{2}. Therefore h^{−1}(γ) = F (a_{0}, …, a_{n}) is in N_{i}^{α}.

Since S is stationary and H : S → H(ω_{2}) is regressive, there is a stationary set S^{∗} ⊆ S and a fixed N such that for all N_{i}^{α} in S^{∗}, H(N_{i}^{α}) = N. The set S^{∗}, being stationary, must have size ω_{2}. So there are distinct α and β such that for some i in d_{α} and j in d_{β}, N_{i}^{α} and N_{j}^{β} are in S^{∗}. Then N = N_{i_{α}}^{α} = N_{i_{β}}^{β}. So N ∩ ω_{2} is in C_{α} ∩ C_{β}, which is a contradiction. □

Abraham and Shelah [1] asked the following question: Assume that A is a stationary subset of ω_{2} ∩ cof(ω_{1}). Does there exist an ω_{1}-distributive forcing poset which adds a club subset to A ∪ cof(ω)? We answer this question in the negative.

Corollary 3.5. Assume that ⟨C_{α} : α ∈ A⟩ is a disjoint club sequence. Let W be an outer model of V with the same ω_{1} and ω_{2}. Then in W, A ∪ cof(ω) does not contain a club subset.

Proof. If A remains stationary in W, then by Lemma 3.3 there is a disjoint club sequence ⟨D_{α} : α ∈ A⟩ in W. By Theorem 3.4 A ∪ cof(ω) does not contain a club in W. □

4. Martin’s Maximum

In this section we prove that Martin’s Maximum implies there exists a disjoint club sequence on ω_{2}. We apply MM to the poset for adding a Cohen real and then forcing a continuous ω_{1}-chain through P_{ω_{1}}(ω_{2}) \ V.

Theorem 4.1 (Krueger). Martin’s Maximum implies there exists a disjoint club sequence on ω_{2}.

We will use the following theorem from [1].

Theorem 4.2. Suppose ℙ is ω_{1}-c.c. and adds a real. Then ℙ forces that (P_{ω_{1}}(ω_{2})\ V) is stationary in P_{ω_{1}}(ω_{2}).

Note: Gitik [6] proved that the conclusion of Theorem 4.2 holds for any outer model of V which contains a new real and computes the same ω_{1}.

Suppose that S is a stationary subset of P_{ω_{1}}(ω_{2}). Following [3] we define a forcing poset ℙ(S) which adds a continuous ω_{1}-chain through S. A condition in ℙ(S) is a countable increasing and continuous sequence ⟨a_{i} : i ≤ β⟩ of elements from S, where for each i < β, a_{i} ∩ ω_{1} < a_{i+1} ∩ ω_{1}. The ordering on ℙ(S) is by extension of sequences.

Proposition 4.3. If S ⊆ P_{ω_{1}}(ω_{2}) is stationary then ℙ(S) is ω-distributive.

Proof. Suppose p forces ḟ : ω → On. Let θ ≫ ω_{2} be a regular cardinal such that ḟ is in H(θ). Since S is stationary, we can fix a countable elementary substructure N of the model

⟨H(θ), ∈, S, ℙ(S), p, ḟ⟩

such that N ∩ ω_{2} is in S. Let ⟨D_{n} : n < ω⟩ be an enumeration of all the dense subsets of ℙ(S) in N. Inductively define a decreasing sequence ⟨p_{n} : n < ω⟩ of elements of N ∩ℙ such that p_{0} = p and p_{n+1} is a refinement of p_{n} in D_{n} ∩N. Write ⋃{p_{n} : n < ω} = ⟨b_{i} : i < γ⟩. Clearly ⋃{b_{i} : i < γ} = N ∩ ω_{2}. Since N ∩ ω_{2} is in S, the sequence ⟨b_{i} : i < γ⟩ ∪ {⟨γ, N ∩ ω_{2}⟩} is a condition below p which decides ḟ(n) for all n < ω. □

Theorem 4.4. Suppose ℙ is an ω_{1}-c.c. forcing poset which adds a real. Let Ṡ be a name such that ℙ forces Ṡ = (P_{ω_{1}}(ω_{2}) \ V). Then ℙ ∗ ℙ(Ṡ) preserves stationary subsets of ω_{1}.

Proof. By Theorem 4.2 and Proposition 4.3, the poset ℙ ∗ ℙ(Ṡ) preserves ω_{1}. Let A be a stationary subset of ω_{1} in V. Suppose p ∗ q̇ is a condition in ℙ ∗ ℙ(Ṡ) which forces Ċ is a club subset of ω_{1}.

Let G be a generic filter for ℙ over V which contains p. In V [G] fix a regular cardinal θ ≫ ω_{2} and let

A = ⟨H(θ), ∈, A, S, q, Ċ⟩.

Fix a Skolem function F : H(θ)^{<ω} → H(θ) for A. Define F^{∗} : ω_{2}^{<ω} → P_{ω_{1}}(ω_{2}) by letting

F^{∗}(α_{0}, …, α_{n}) = cl_{F} ({α_{0}, …, α_{n}}) ∩ ω_{2}.

Since ℙ is ω_{1}-c.c. there is a function H : ω_{2}^{<ω} → P_{ω_{1}}(ω_{2}) in V such that for all α⃗ in ω_{2}^{<ω}, F^{∗}(α⃗) ⊆ H(α⃗). Let Z^{∗} be the stationary set of α in ω_{2} ∩ cof(ω) closed under H.

Working in V, since A is stationary we can fix for each α in Z^{∗} a countable cofinal set M_{α} ⊆ α closed under H with M_{α} ∩ ω_{1} in A. By Fodor’s Lemma there is Z ⊆ Z^{∗} stationary and δ in A such that for all α in Z, M_{α} ∩ω_{1} = δ. Fix a sequence ⟨ξ_{s}, Z_{s} : s ∈ ^{<ω}2⟩ satisfying conditions (1)–(4) of Lemma 2.6.

Let f : ω → 2 be a function in V [G] \ V. For each n < ω let M_{n} denote cl_{H}(δ ∪ {ξ_{f↾m} : m ≤ n}). Define M = ⋃{M_{n} : n < ω}. Note that M is closed under H and hence it is closed under F^{∗}. Therefore N = cl_{F} (M) is an elementary substructure of A such that N ∩ ω_{2} = M.

As in the proof of Theorem 2.7, for all n < ω, if α is in Z_{f↾n} then M_{n} ⊆ M_{α}. Note that M ∩ ω_{1} = δ. For if γ is in M ∩ ω_{1}, there is n < ω such that γ is in M_{n}. Fix α in Z_{f↾n}. Then γ is in M_{α} ∩ ω_{1} = δ.

We prove that M is not in V by showing how to compute f by induction from M. Suppose f ↾ n is known. Fix j < 2 such that f(n) ≠ j. We claim that ξ_{(f↾n) j} is not in M. Otherwise there is k > n such that ξ_{(f↾n) j} is in M_{k}. Fix α in Z_{f↾k}. Then ξ_{(f↾n) j} is in M_{α}. But α is in Z_{f↾(n+1)}, contradicting Lemma 2.6(4). So f(n) is the unique i < 2 such that ξ_{(f↾n) i} is in M. This completes the definition of f from M. Since f is not in V, neither is M.

Let ⟨D_{n} : n < ω⟩ enumerate the dense subsets of ℙ(S) lying in N. Inductively define a decreasing sequence ⟨q_{n} : n < ω⟩ in N ∩ℙ(S) such that q_{0} = q and q_{n+1} is in D_{n} ∩ N. Write ⋃{q_{n} : n < ω} = ⟨b_{i} : i < γ⟩. Clearly ⋃{b_{i} : i < γ} = N ∩ ω_{2} = M, and since M is not in V, r = ⟨b_{i} : i < γ⟩ ∪ {⟨γ, M⟩} is a condition in ℙ(S). By an easy density argument, r forces that N ∩ ω_{1} = δ is a limit point of Ċ, and hence is in Ċ. Let ṙ be a name for r. Then p ∗ ṙ ≤ p ∗ q̇ and p ∗ ṙ forces δ is in A ∩ Ċ. □

Now we are ready to prove that MM implies there exists a disjoint club sequence on ω_{2}.

Proof of Theorem 4.1. Assume Martin’s Maximum. Inductively define A and ⟨C_{α} : α ∈ A⟩ as follows. Suppose α is in ω_{2} ∩ cof(ω_{1}) and A ∩ α and ⟨C_{β} : β ∈ A ∩ α⟩ are defined. Let α be in A iff the set ⋃{C_{β} : β ∈ A ∩ α} is non-stationary in P_{ω_{1}}(α). If α is in A then choose a club set C_{α} ⊆ P_{ω_{1}}(α) with size ω_{1} which is disjoint from this union.

This completes the definition. We prove that A is stationary. Then clearly ⟨C_{α} : α ∈ A⟩ is a disjoint club sequence. Fix a club set C ⊆ ω_{2}.

Let Add denote the forcing poset for adding a single Cohen real with finite conditions and let Ṡ be an Add-name for the set (P_{ω_{1}}(ω_{2}) \ V). By Theorem 4.4 the poset Add ∗ ℙ(Ṡ) preserves stationary subsets of ω_{1}. We will apply Martin’s Maximum to this poset after choosing a suitable collection of dense sets.

For each α < ω_{2} fix a surjection f_{α} : ω_{1} → α. If β is in A enumerate C_{β} as ⟨a_{i}^{β} : i < ω_{1}⟩. For every quadruple i, j, k, l of countable ordinals let D(i, j, k, l) denote the set of conditions p ∗ q̇ such that:

(1) p forces that i and j are in the domain of q̇, and for some β_{i} and β_{j}, p forces β_{i} = sup(q̇(i)) and β_{j} = sup(q̇(j)),

(2) there is some ζ < ω_{1} such that p forces ζ is the least element in dom(q̇) such that f_{β_{i}}(j) ∈ q̇(ζ),

(3) there is ξ in C larger than β_{i} and β_{j} such that p forces ξ is the supremum of the maximal set in q̇,

(4) if f_{β_{j}}(k) = γ is in A, then there is z such that p forces z is in the symmetric difference q̇(i)△a_{l}^{γ}.

It is routine to check that D(i, j, k, l) is dense.

Let G ∗ H be a filter on Add ∗ ℙ(Ṡ) intersecting each D(i, j, k, l). For i < ω_{1} define a_{i} as the set of β for which there exists some p ∗ q̇ in G ∗ H such that p forces i ∈ dom(q̇) and p forces β is in q̇(i). The definition of the dense sets implies that ⟨a_{i} : i < ω_{1}⟩ is increasing, continuous, and cofinal in P_{ω_{1}}(α) for some α in C ∩cof(ω_{1}). By (4), for each γ in A∩α, {a_{i} : i < ω_{1}} is disjoint from C_{γ}. Therefore ⋃{C_{γ} : γ ∈ A ∩ α} is non-stationary in P_{ω_{1}}(α), hence by the definition of A, α is in A ∩ C. So A is stationary. □

5. The Equiconsistency Result

We now prove Theorem 0.1 establishing the consistency strength of each of the following statements to be exactly a Mahlo cardinal: (1) There does not exist a thin stationary subset of P_{ω_{1}}(ω_{2}). (2) There exists a disjoint club sequence on ω_{2}. (3) There exists a fat stationary set S ⊆ ω_{2} such that any forcing poset which preserves ω_{1} and ω_{2} does not add a club subset to S.

By [5] if there exists a thin stationary subset of P_{ω_{1}}(ω_{2}) then for any fat stationary set S ⊆ ω_{2}, there is a forcing poset which preserves cardinals and adds a club subset to S. So (2) and (3) both imply (1), which in turn implies there is no special Aronszajn tree on ω_{2}. So ω_{2} is Mahlo in L by [8].

In the other direction assume that κ is a Mahlo cardinal. We will define a revised countable support iteration which collapses κ to become ω_{2} and adds a disjoint club sequence on ω_{2}. At individual stages of the iteration we force with either a collapse forcing or the poset Add ∗ ℙ(Ṡ) from the previous section. To ensure that ω_{1} is not collapsed we verify that Add∗ℙ(Ṡ) satisfies an iterable condition known as the 𝕀-universal property. Our description of this construction is self-contained, except for the proof of Theorem 5.9 which summarizes the relevant properties of the RCS iteration. For more information on such iterations and the 𝕀-universal property see [10].

Definition 5.1. A pair ⟨T, I⟩ is a tagged tree if:

(1) T ⊆ ^{<ω}On is a tree such that each η in T has at least one successor,

(2) I : T → V is a partial function such that each I(η) is an ideal on some set X_{η} and for each η in the domain of I, the set {α : η α ∈ T } is in (I(η))^{+},

(3) for each cofinal branch b of T, there are infinitely many n < ω such that b ↾ n is in the domain of I.

If η is in the domain of I, we say that η is a splitting point of T. It follows from (1) and (3) that for every η in T there is η ◁ ν which is a splitting point.

Definition 5.2. Let 𝕀 be a family of ideals and ⟨T, I⟩ a tagged tree. Then ⟨T, I⟩ is an 𝕀-tree if for each splitting point η in T, I(η) is in 𝕀.

Suppose T ⊆ ^{<ω}On is a tree. If η is in T, let T^{[η]} denote the tree {ν ∈ T : ν ⊴ η or η ⊴ ν}. A set J ⊆ T is called a front if for distinct nodes in J, neither is an initial segment of the other, and for any cofinal branch b of T there is η in J which is an initial segment of b.

Definition 5.3. Suppose ⟨T, I⟩ is tagged tree. Let θ be a regular cardinal such that ⟨T, I⟩ is in H(θ), and let <_{θ} be a well-ordering of H(θ). A sequence ⟨N_{η} : η ∈ T ⟩ is a tree of models for θ provided that:

(1) each N_{η} is a countable elementary substructure of ⟨H(θ), ∈, <_{θ}, ⟨T, I⟩⟩,

(2) if η ◁ ν in T, then N_{η} ≺ N_{ν},

(3) for each η in T, η is in N_{η}.

Definition 5.4. Suppose ⟨T, I⟩ is an 𝕀-tree, and θ is a regular cardinal such that H(θ) contains ⟨T, I⟩ and 𝕀. A sequence ⟨N_{η} : η ∈ T ⟩ is an 𝕀-suitable tree of models for θ if it is a tree of models for θ and for every η in T and I in 𝕀 ∩ N_{η}, the set

{ν ∈ T^{[η]} : ν is a splitting point and I(ν) = I}

contains a front in T^{[η]}.

Definition 5.5. Let ⟨T, I⟩, 𝕀, and θ be as in Definition 5.4. A sequence ⟨N_{η} : η ∈ T ⟩ is an ω_{1}-strictly 𝕀-suitable tree of models for θ if it is an 𝕀-suitable tree of models for θ and there exists δ < ω_{1} such that for all η in T, N_{η} ∩ ω_{1} = δ.

If ⟨N_{η} : η ∈ T ⟩ is a tree of models and b is a cofinal branch of T, write N_{b} for the set ⋃{N_{b↾n} : n < ω}. Note that if ⟨N_{η} : η ∈ T ⟩ is an ω_{1}-strictly 𝕀-suitable tree of models for θ, then for any cofinal branch b of T, N_{b} ∩ ω_{1} = N_{⟨⟩} ∩ ω_{1}.

Lemma 5.6. Let ⟨T, I⟩, 𝕀, and θ be as in Definition 5.4, and let ⟨N_{η} : η ∈ T ⟩ be an ω_{1}-strictly 𝕀-suitable tree of models for θ. Suppose η ◁ ν in T and (N_{ν} ∩ ω_{2}) \ N_{η} is non-empty. Let γ be the minimum element of (N_{ν}∩ω_{2})\N_{η}. Then γ ≥ sup(N_{η}∩ω_{2}).

Proof. Otherwise there is β in N_{η} ∩ ω_{2} such that γ < β. By elementarity, there is a surjection f : ω_{1} → β in N_{η}. So f^{−1}(γ) ∈ N_{ν} ∩ω_{1} = N_{η} ∩ω_{1}. Hence f(f^{−1}(γ)) = γ is in N_{η}, which is a contradiction. □

Let 𝕀 be a family of ideals. We say that 𝕀 is restriction-closed if for all I in 𝕀, for any set A in I^{+}, the ideal I ↾ A is in 𝕀. If μ is a regular uncountable cardinal, we say that 𝕀 is μ-complete if each ideal in 𝕀 is μ-complete.

Definition 5.7. Suppose that 𝕀 is a non-empty restriction-closed ω_{2}-complete family of ideals and let ℙ be a forcing poset. Then ℙ satisfies the 𝕀-universal property if for all sufficiently large regular cardinals θ with 𝕀 in H(θ), if ⟨N_{η} : η ∈ T ⟩ is an ω_{1}-strictly 𝕀-suitable tree of models for θ, then for all p in N_{⟨⟩} ∩ ℙ there is q ≤ p such that q forces there is a cofinal branch b of T such that N_{b}[Ġ] ∩ ω_{1} = N_{⟨⟩} ∩ ω_{1}.

Definition 5.7 is Shelah’s characterization of the 𝕀-universal property given in [10] Chapter XV 2.11, 2.12, and 2.13. Note that in the definition, q is semigeneric for N_{⟨⟩}. In 2.12 Shelah proves that there are stationarily many structures N for which N = N_{⟨⟩} for some ω_{1}-strictly 𝕀-suitable tree of models ⟨N_{η} : η ∈ T ⟩. So by standard arguments if ℙ satisfies the 𝕀-universal property then ℙ preserves ω_{1} and preserves stationary subsets of ω_{1}. Note that any semiproper forcing poset satisfies the 𝕀-universal property.

Theorem 5.8. Let 𝕀 be the family of ideals of the form NS_{ω_{2}} ↾ A, where A is a stationary subset of ω_{2} ∩ cof(ω). Let Ṡ be an Add-name for the set (P_{ω_{1}}(ω_{2}) \ V). Then Add ∗ ℙ(Ṡ) satisfies the 𝕀-universal property.

Proof. Fix a regular cardinal θ ≫ ω_{2} and let ⟨N_{η} : η ∈ T ⟩ be an ω_{1}-strictly 𝕀suitable tree of models for θ. Let p ∗ q̇ be a condition in (Add ∗ ℙ(Ṡ)) ∩ N_{⟨⟩}. We find a refinement of p ∗ q̇ which forces there is a cofinal branch b of T such that N_{b}[Ġ ∗ Ḣ] ∩ ω_{1} = N_{⟨⟩} ∩ ω_{1}.

We use an argument similar to the proof of Lemma 2.6 to define a sequence ⟨η_{s}, ξ_{s} : s ∈ ^{<ω}2⟩ satisfying:

(1) each η_{s} is in T, each ξ_{s} is in N_{η_{s}} ∩ ω_{2}, and s ◁ t implies η_{s} ◁ η_{t},

(2) if s 0 ⊴ t then ξ_{s 1} is not in N_{η_{t}}, and if s 1 ⊴ u then ξ_{s 0} is not in N_{η_{u}}. Let η_{⟨⟩} = ⟨⟩ and ξ_{⟨⟩} = 0. Suppose η_{s} is defined. Choose a splitting point ν_{s} in

T above η_{s}. Let Z denote the set of α < ω_{2} such that ν_{s} α is in T. Since ν_{s} is a splitting point, by the definition of 𝕀 the set Z is a stationary subset of ω_{2} ∩ cof(ω). For each α in Z, α is in N_{(ν_{s} α)} and has cofinality ω, so N_{(ν_{s} α)} ∩ α is a countable cofinal subset of α. Define X_{s} as the set of ξ in ω_{2} such that the set

{α ∈ Z : ξ ∈ N_{(ν_{s} α)} ∩ α}

is stationary. An easy argument using Fodor’s Lemma shows that X_{s} is unbounded in ω_{2}. For all large enough α in Z, the set (X_{s} \ sup(N_{ν_{s}} ∩ ω_{2})) ∩ α has size ω_{1}. So there is a stationary set Z′_{1} ⊆ Z and an ordinal ξ_{s 0} in X_{s} such that ξ_{s 0} is larger than sup(N_{ν_{s}} ∩ ω_{2}) and for all α in Z′_{1}, ξ_{s 0} is not in N_{(ν_{s} α)} ∩ α. Let Z′_{0} be the stationary set of α in Z such that ξ_{s 0} is in N_{(ν_{s} α)} ∩ α. Now define Y_{s} as the set of ξ in ω_{2} such that the set

{α ∈ Z′_{1} : ξ ∈ N_{(ν_{s} α)} ∩ α}

is stationary. Again we can find Z_{0} ⊆ Z′_{0} stationary and ξ_{s 1} in Y_{s} such that ξ_{s 1} is larger than sup(N_{ν_{s}} ∩ ω_{2}) and for all α in Z_{0}, ξ_{s 1} is not in N_{(ν_{s} α)} ∩ α. Let Z_{1} be the stationary set of α in Z′_{1} such that ξ_{s 1} is in N_{(ν_{s} α)} ∩ α.

Now define η_{s 0} to be equal to ν_{s} α for some α in Z_{0} larger than ξ_{s 1}, and define η_{s 1} to be ν_{s} β for some β in Z_{1} larger than ξ_{s 0}. By definition ξ_{s 0} is in N_{η_{s 0}} and ξ_{s 1} is in N_{η_{s 1}}.

We claim that if η_{s 0} ⊴ ν in T, then ξ_{s 1} is not in N_{ν}. Since α is in Z_{0}, ξ_{s 1} is not in N_{(η_{s 0})} ∩ α. But ξ_{s 1} < α, so ξ_{s 1} is not in N_{(η_{s 0})}. By Lemma 5.6 the minimum element of N_{ν} ∩ ω_{2} which is not in N_{(η_{s 0})}, if such an ordinal exists, is at least sup(N_{(η_{s 0})} ∩ ω_{2}) ≥ α > ξ_{s 1}. So ξ_{s 1} is not in N_{ν}. Similarly if η_{s 1} ⊴ ν in T, then ξ_{s 0} is not in N_{ν}. This completes the definition. Conditions (1) and (2) are now easily verified.

Since ℙ is ω_{1}-c.c., the condition p itself is generic for each N_{η}. Let G be a generic filter for Add over V which contains p. Then for all η in T, N_{η}[G] ∩ ω_{2} = N_{η} ∩ ω_{2}.

So for any cofinal branch b of T in V [G], N_{b}[G] ∩ ω_{2} = ⋃{N_{b↾n} ∩ ω_{2} : n < ω}; in particular, N_{b}[G] ∩ ω_{1} = N_{⟨⟩} ∩ ω_{1}.

Let f : ω → 2 be a function in V [G] \ V. Define b_{f} = ⋃{η_{f↾n} : n < ω}. We prove that N_{b_{f}} ∩ ω_{2} is not in V by showing how to define f inductively from this set. Suppose f ↾ n is known. Fix j < 2 such that f(n) ≠ j. We claim that ξ^{∗} = ξ_{(f↾n) j} is not in N_{b_{f}} ∩ ω_{2}. Otherwise there is k > n such that ξ^{∗} is in N_{η_{f↾k}}. But f ↾ (n + 1) ⊴ f ↾ k. So by condition (2), ξ^{∗} is not in N_{η_{f↾k}}, which is a contradiction. So f(n) is the unique i < 2 such that ξ_{(f↾n) i} is in N_{b_{f}} ∩ ω_{2}.

Let ⟨D_{n} : n < ω⟩ enumerate all the dense subsets of ℙ(S) lying in N_{b_{f}} [G]. Inductively define a sequence ⟨q_{n} : n < ω⟩ by letting q_{0} = q and choosing q_{n+1} to be a refinement of q in D_{n} ∩ N_{b_{f}} [G]. Let ⟨b_{i} : i < γ⟩ = ⋃{q_{n} : n < ω}. Clearly ⋃{b_{i} : i < γ} = N_{b_{f}} ∩ω_{2}. Since N_{b_{f}} ∩ω_{2} is not in V, r = ⟨b_{i} : i < γ⟩ (N_{b_{f}} ∩ω_{2}) is a condition in ℙ(S) below q and r is generic for N_{b_{f}} [G]. So r forces N_{b_{f}} [G][Ḣ] ∩ ω_{1} = N_{b_{f}} [G] ∩ ω_{1} = N_{⟨⟩} ∩ ω_{1}. Let ṙ be a name for r. Then p ∗ ṙ ≤ p ∗ q̇ is as required. □

We state without proof the facts concerning RCS iterations which we shall use. These facts follow immediately from [10] Chapter XI 1.13 and Chapter XV 4.15.

Theorem 5.9. Suppose ⟨ℙ_{i}, ℚ_{j} : i ≤ α, j < α⟩ is an RCS iteration. Then ℙ_{α} preserves ω_{1} if the iteration satisfies the following properties:

(1) for each i < α there is n < ω such that ℙ_{i+n} ⊩ |ℙ_{i}| ≤ ω_{1},

(2) for each i < α there is an uncountable regular cardinal κ_{i} and a ℙ_{i}-name 𝕀̇_{i} such that ℙ_{i} is κ_{i}-c.c. and ℙ_{i} forces 𝕀̇_{i} is a non-empty restriction-closed κ_{i}-complete family of ideals such that ℚ̇_{i} satisfies the 𝕀̇_{i}-universal property.

Theorem 5.10. Let α be a strongly inaccessible cardinal. Suppose that ⟨ℙ_{i}, ℚ_{j} : i ≤ α, j < α⟩ is a revised countable support iteration such that ℙ_{α} preserves ω_{1} and for all i < α, |ℙ_{i}| < α. Then ℙ_{α} is α-c.c.

Suppose κ is a Mahlo cardinal and let A be the stationary set of strongly inaccessible cardinals below κ. Define an RCS iteration ⟨ℙ_{i}, ℚ̇_{j} : i ≤ κ, j < κ⟩ by recursion as follows. Our recursion hypotheses will include that each ℙ_{α} preserves ω_{1}, and is α-c.c. if α is in A.

Suppose ℙ_{α} is defined. If α is not in A then let ℚ̇_{α} be a name for Coll(ω_{1}, |ℙ_{α}|). Suppose α is in A. By the recursion hypotheses ℙ_{α} forces α = ω_{2}. Let ℚ̇_{α} be a name for the poset Add ∗ ℙ(Ṡ).

If α is not in A then choose some regular cardinal κ_{α} larger than |ℙ_{α}|, and let 𝕀̇_{α} be a name for some non-empty restriction-closed κ_{α}-complete family of ideals on κ_{α}. Then ℙ_{α} is κ_{α}-c.c., and since ℚ̇_{α} is proper, ℙ_{α} forces ℚ̇_{α} satisfies the 𝕀̇_{α}- universal property. Suppose α is in A. Then let α = κ_{α} and define 𝕀̇_{α} as a name for the family of ideals on ω_{2} as described in Theorem 5.8. Then ℙ_{α} is κ_{α}-c.c. and forces that ℚ̇_{α} satisfies the 𝕀̇_{α}-universal property.

Suppose β ≤ κ is a limit ordinal and ℙ_{α} is defined for all α < β. Define ℙ_{β} as the revised countable support limit of ⟨ℙ_{α} : α < β⟩. By Theorem 5.9 and the recursion hypotheses, ℙ_{β} preserves ω_{1}. Hence if β is in A ∪ {κ}, then ℙ_{β} is β-c.c. by Theorem 5.10.

This completes the definition. Let G be generic for ℙ_{κ}. The poset ℙ_{κ} is κ-c.c. and preserves ω_{1}, so in V [G] we have that κ = ω_{2} and A is a stationary subset of ω_{2} ∩ cof(ω_{1}). For each α in A let C_{α} be the club on P_{ω_{1}}(α) introduced by ℚ_{α}. If α < β are in A, then C_{α} and C_{β} are disjoint since C_{β} is disjoint from V [G ↾ β]. So ⟨C_{α} : α ∈ A⟩ is a disjoint club sequence on ω_{2} in V [G].

We conclude the paper with several questions.

(1) Assuming Martin’s Maximum, the poset Add ∗ ℙ(Ṡ) is semiproper. Is this poset semiproper in general?

(2) Is it consistent that there exists a stationary set A ⊆ ω_{2} ∩ cof(ω_{1}) such that neither A ∪ cof(ω) nor ω_{2} \ A can acquire a club subset in an ω_{1} and ω_{2} preserving extension?

(3) To what extent can the results of this paper be extended to cardinals greater than ω_{2}? For example, is it consistent that there is a fat stationary subset of ω_{3} which cannot acquire a club subset by any forcing poset which preserves ω_{1}, ω_{2}, and ω_{3}?

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Kurt Gödel Research Center for Mathematical Logic, University of Vienna

  E-mail address: sdf@logic.univie.ac.at
  URL: http://www.logic.univie.ac.at/∼sdf

Kurt Gödel Research Center for Mathematical Logic, University of Vienna

  E-mail address: jkrueger@logic.univie.ac.at
  URL: http://www.logic.univie.ac.at/∼jkrueger